Autonomous Drone Flight Kinematics

Autonomous quadcopter drones navigate urban delivery corridors by evaluating onboard sensor feedback and tracking pre-computed 3D parametric spline trajectories. Aerospace and computer engineering teams decouple the drone's instantaneous flight speed from its 3D directional heading vector using the principal unit tangent vector $\hat{T}(t)$, which feeds directly into flight control algorithms for gyroscopic attitude correction.

An autonomous quadcopter executes an ascending spiral corridor inspection with 3D position vector (in meters) given as a function of time $t$ (in seconds):

1. Calculate the instantaneous velocity vector $\vec{v}(t) = \vec{r}'(t)$ and the acceleration vector $\vec{a}(t) = \vec{r}''(t)$ of the quadcopter.
2. Determine the scalar flight speed $\|\vec{v}(t)\|$ of the drone and verify whether it travels at constant speed.
3. Compute the principal unit tangent vector $\hat{T}(t)$ defining the drone's directional flight heading.
4. Evaluate the velocity $\vec{v}(\pi)$, acceleration $\vec{a}(\pi)$, and unit tangent heading $\hat{T}(\pi)$ at time $t = \pi\text{ seconds}$.